Nonlocal approximation of minimal surfaces: optimal estimates from stability

TL;DR

This paper develops a new nonlocal approximation for minimal surfaces, providing optimal estimates on stability, regularity, and sheet separation, and characterizes stable s-minimal hypersurfaces in high dimensions.

Contribution

It introduces a nonlocal minimal surface approximation with uniform estimates as s approaches 1 and characterizes stable s-minimal hypersurfaces in 4, extending prior methods.

Findings

Robust $C^{2,}$ estimates for stable $s$-minimal surfaces as $srrow 1$

Optimal sheet separation of order $(1-s)^{1/2}$ between components

Hyperplanes are the only stable $s$-minimal hypersurfaces in 4 for $s$ close to 1

Abstract

Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if $\partial E$ is a stable $s$-minimal surface in $B_1\subset \mathbb R^3$ then: - $\partial E\cap B_{1/2}$ enjoys a $C^{2,\alpha}$ estimate that is robust as $s\uparrow 1$ (i.e. uniform in $s$); - the distance between different connected components of~$\partial E\cap B_{1/2}$ must be at least of order~$(1-s)^{\frac 1 2}$ (optimal sheet separation estimate); - interactions between multiple sheets at distances of order $(1-s)^{\frac 1 2}$ are described by the D\'avila--del Pino--Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable $s$-minimal hypersurfaces in $\mathbb R^4$, for $s\in(0,1)$ sufficiently close to $1$. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen-Cahn).